Add, Subtract & Check Fraction Amounts

c203-pre-1. Find 1/3 + 1/4. Show how to make equal-sized units before adding.

Answer: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The addends are renamed in twelfths of the same whole.

Look for:

• 0 points: Adds numerators and denominators to get 2/7. No correct supporting model or explanation is present.
• 1 point: Gets 7/12 or writes valid equivalent fractions but omits a complete common-unit explanation.
• 2 points: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The addends are renamed in twelfths of the same whole.

Teacher response: Watch for: Adds numerators and denominators to get 2/7. Respond: Partition one whole into twelve parts and locate both addends within it.

c203-pre-2. A bottle holds 5/6 liter. After 1/3 liter is poured out, how much remains? Use equivalent fractions and include units.

Answer: 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 liter. Both amounts are compared in sixths of a liter.

Look for:

• 0 points: Subtracts denominators or removes one sixth instead of one third. No correct supporting model or explanation is present.
• 1 point: Gets 1/2 liter or correctly renames 1/3 as 2/6 but leaves the explanation incomplete.
• 2 points: 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 liter. Both amounts are compared in sixths of a liter.

Teacher response: Watch for: Subtracts denominators or removes one sixth instead of one third. Respond: Mark one third as two sixths on the bottle before crossing it out.

c203-pre-3. Find 2 1/4 − 3/4. Show the exchange of one whole before subtracting.

Answer: 2 1/4 = 1 5/4, so 1 5/4 − 3/4 = 1 2/4 = 1 1/2.

Look for:

• 0 points: Subtracts the smaller numerator from the larger without preserving the subtraction order. No correct supporting model or explanation is present.
• 1 point: Gives 1 1/2 but omits the valid exchange, or shows 1 5/4 with a later calculation error.
• 2 points: 2 1/4 = 1 5/4, so 1 5/4 − 3/4 = 1 2/4 = 1 1/2.

Teacher response: Watch for: Subtracts the smaller numerator from the larger without preserving the subtraction order. Respond: Exchange one whole for four fourths and combine those with the existing fourth.

c203-pre-4. A student rewrites 3 1/5 as 2 5/5 to prepare for subtraction. Correct the rewrite and explain the missing amount.

Answer: 3 1/5 = 2 6/5. One exchanged whole supplies 5/5, and the existing 1/5 must still be included; 2 5/5 loses 1/5.

Look for:

• 0 points: Drops the existing fractional part when exchanging a whole. No correct supporting model or explanation is present.
• 1 point: Corrects the rewrite to 2 6/5 but does not explain the lost fifth.
• 2 points: 3 1/5 = 2 6/5. One exchanged whole supplies 5/5, and the existing 1/5 must still be included; 2 5/5 loses 1/5.

Teacher response: Watch for: Drops the existing fractional part when exchanging a whole. Respond: Keep the original fifth visible while adding the five fifths from the exchanged whole.

c203-pre-5. Someone claims 2/3 + 3/4 = 5/7. Reject or accept this using an estimate before finding the exact sum.

Answer: Reject it: both addends exceed 1/2, so the sum exceeds 1. The exact sum is 8/12 + 9/12 = 17/12 = 1 5/12, while 5/7 is below 1.

Look for:

• 0 points: Accepts an added-denominators answer even though it is too small. No correct supporting model or explanation is present.
• 1 point: Gives a valid estimate or exact sum, but does not use both to evaluate the claim.
• 2 points: Reject it: both addends exceed 1/2, so the sum exceeds 1. The exact sum is 8/12 + 9/12 = 17/12 = 1 5/12, while 5/7 is below 1.

Teacher response: Watch for: Accepts an added-denominators answer even though it is too small. Respond: Place both addends beyond 1/2 and compare their combined length with one whole.

c203-pre-6. A trail is 1 1/2 km long. You walk 2/3 km. Is 1/6 km a reasonable remaining distance? Find the remainder and check by addition.

Answer: No. 1 1/2 − 2/3 = 9/6 − 4/6 = 5/6 km; 5/6 + 2/3 = 9/6 = 1 1/2 km.

Look for:

• 0 points: Ignores the whole-number part when finding the remaining distance. No correct supporting model or explanation is present.
• 1 point: Finds 5/6 km but omits the addition check, or rejects 1/6 with a valid partial model.
• 2 points: No. 1 1/2 − 2/3 = 9/6 − 4/6 = 5/6 km; 5/6 + 2/3 = 9/6 = 1 1/2 km.

Teacher response: Watch for: Ignores the whole-number part when finding the remaining distance. Respond: Draw the entire 1 1/2 km first and show the walked and remaining parts joining to it.

c203-post-1. A jug receives 2/5 liter and then 1/2 liter. Find the combined amount and explain the equal-sized parts you used.

Answer: 2/5 + 1/2 = 4/10 + 5/10 = 9/10 liter. Both amounts are renamed as tenths of a liter.

Look for:

• 0 points: Adds unlike part counts without renaming them in a shared unit. No correct supporting model or explanation is present.
• 1 point: Gives 9/10 liter or correct tenths but leaves the common-unit explanation incomplete.
• 2 points: 2/5 + 1/2 = 4/10 + 5/10 = 9/10 liter. Both amounts are renamed as tenths of a liter.

Teacher response: Watch for: Adds unlike part counts without renaming them in a shared unit. Respond: Draw fifths and halves on equal liter bars, then subdivide both into tenths.

c203-post-2. A rope is 7/8 meter long. A 1/4-meter piece is cut off. Show the remaining length on a bar and as an equation.

Answer: 7/8 − 1/4 = 7/8 − 2/8 = 5/8 meter. The bar removes two eighths from seven eighths.

Look for:

• 0 points: Removes one eighth for a quarter or subtracts the denominators. No correct supporting model or explanation is present.
• 1 point: Gives 5/8 meter but does not align the bar with the equivalent-fraction equation.
• 2 points: 7/8 − 1/4 = 7/8 − 2/8 = 5/8 meter. The bar removes two eighths from seven eighths.

Teacher response: Watch for: Removes one eighth for a quarter or subtracts the denominators. Respond: Fold the quarter in two to show why it occupies two eighths.

c203-post-3. There are 4 1/3 cups of flour. A recipe uses 1 2/3 cups. Find the amount left and show a value-preserving exchange.

Answer: 4 1/3 = 3 4/3; subtracting 1 2/3 leaves 2 2/3 cups.

Look for:

• 0 points: Subtracts fractional numerators in reverse order when the first is smaller. No correct supporting model or explanation is present.
• 1 point: Finds 2 2/3 cups but omits the exchange, or correctly writes 3 4/3 before a later error.
• 2 points: 4 1/3 = 3 4/3; subtracting 1 2/3 leaves 2 2/3 cups.

Teacher response: Watch for: Subtracts fractional numerators in reverse order when the first is smaller. Respond: Exchange one of the four whole cups for three thirds and add the original third.

c203-post-4. Which is equal to 5 2/6: 4 6/6, 4 8/6, or 5 8/6? Explain by exchanging exactly one whole.

Answer: 4 8/6 is equal. The exchanged whole adds 6/6 to the existing 2/6 and reduces the whole count from five to four.

Look for:

• 0 points: Adds fractional parts from an exchange without adjusting the whole count. No correct supporting model or explanation is present.
• 1 point: Selects 4 8/6 but does not explain both the reduced whole count and the eight sixths.
• 2 points: 4 8/6 is equal. The exchanged whole adds 6/6 to the existing 2/6 and reduces the whole count from five to four.

Teacher response: Watch for: Adds fractional parts from an exchange without adjusting the whole count. Respond: Draw five wholes and two sixths; replace exactly one whole with six sixths.

c203-post-5. Before solving 5/8 + 7/10, decide whether its sum is below or above 1. Then calculate and explain why 12/18 cannot be correct.

Answer: It is above 1 because each addend exceeds 1/2. The sum is 25/40 + 28/40 = 53/40 = 1 13/40; 12/18 is below 1.

Look for:

• 0 points: Accepts a result below 1 for the sum of two amounts above 1/2. No correct supporting model or explanation is present.
• 1 point: Gives a valid estimate or 53/40, but does not connect both to rejecting 12/18.
• 2 points: It is above 1 because each addend exceeds 1/2. The sum is 25/40 + 28/40 = 53/40 = 1 13/40; 12/18 is below 1.

Teacher response: Watch for: Accepts a result below 1 for the sum of two amounts above 1/2. Respond: Estimate both addends first, and keep that lower bound visible during calculation.

c203-post-6. A tank starts with 2 1/4 liters and ends with 5/6 liter. How much was used? Check that used plus remaining equals the starting amount.

Answer: 2 1/4 − 5/6 = 27/12 − 10/12 = 17/12 = 1 5/12 liters used. Then 17/12 + 10/12 = 27/12 = 2 1/4.

Look for:

• 0 points: Treats the remaining quantity as the amount used or drops the whole liters. No correct supporting model or explanation is present.
• 1 point: Finds 17/12 liters or a correct check equation but omits another requested part.
• 2 points: 2 1/4 − 5/6 = 27/12 − 10/12 = 17/12 = 1 5/12 liters used. Then 17/12 + 10/12 = 27/12 = 2 1/4.

Teacher response: Watch for: Treats the remaining quantity as the amount used or drops the whole liters. Respond: Label start, used and remaining on a single bar, then write their part–whole equation.

c203-step-1. 1. Compose One: Compose and decompose a whole using equal units. Lesson: /lessons/compose-one/

Answer: Can the learner explain why 3/8 + 5/8 has the same unit throughout?

c203-step-2. 2. Mixed Number Market: Compose whole and fractional amounts in a context. Lesson: /lessons/mixed-number-market/

Answer: Does the learner rename a complete group of parts as one whole?

c203-step-3. 3. Common Denominator Kitchen: Create common units before adding unlike fractions. Lesson: /lessons/common-denominator-kitchen/

Answer: Ask what the new denominator means in the model.

c203-step-4. 4. Fraction Difference Trail: Find a missing distance or amount with common units. Lesson: /lessons/fraction-difference-trail/

Answer: Can the learner check the difference by adding it back?

c203-step-5. 5. Mixed Number Repair: Regroup a whole to make a subtraction possible. Lesson: /lessons/mixed-number-repair/

Answer: Does the renamed mixed number preserve its original value?

c203-step-6. 6. Estimate Before Exact: Predict a range, solve, then evaluate the result. Lesson: /lessons/estimate-before-exact/

Answer: Look for a size check that would detect an added-denominators answer.

c203-skill-1. Common units: items 1–2 on both assessments. Evidence: Renames unlike fractions before combining or removing their counts.

Answer: 0–1: reteach. Overlay equal-whole strips to create a shared partition. Name both amounts in that unit before calculating. /interventions/equivalent-factors/. 2–3: guided practice. Use Common Denominator Kitchen and explain why the denominator names a unit, not a count to add. /lessons/common-denominator-kitchen/. 4: extend. Solve a difference with two common denominators and explain why both results agree. /lessons/fraction-difference-trail/. Missing evidence: collect another response before grouping.

c203-skill-2. Regroup mixed numbers: items 3–4 on both assessments. Evidence: Trades one whole for fractional units while preserving total value.

Answer: 0–1: reteach. Exchange one whole strip for all of its equal fractional pieces before subtracting. /interventions/mixed-regrouping/. 2–3: guided practice. Record the mixed number before and after the exchange and verify they are equal. /lessons/mixed-number-repair/. 4: extend. Show two different subtraction strategies and check each by addition. /lessons/mixed-number-market/. Missing evidence: collect another response before grouping.

c203-skill-3. Estimate and check: items 5–6 on both assessments. Evidence: Uses size and context to reject impossible results and confirms exact amounts.

Answer: 0–1: reteach. Place the addends near benchmarks and set a reasonable range before calculating. /interventions/unlike-estimation/. 2–3: guided practice. Compare the exact result with the estimate and with each starting amount. /lessons/estimate-before-exact/. 4: extend. Create an incorrect fraction calculation that a size check can disprove immediately. /lessons/fraction-difference-trail/. Missing evidence: collect another response before grouping.